WendlandC63D.hcl File Reference

Wendland kernel definition (3D version). More...

Macros
#define	_KERNEL_H_INCLUDED_
#define	M_PI   3.14159265359f
#define	iM_PI   0.318309886f

Functions
const float	kernelW (const float q)
	The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
const float	kernelF (const float q)
	The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).
const float	kernelS_P (const float q)
	An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.
const float	_Omega (const float a, const float b)
	Helper function to compute the solid angle of a rectangular patch, with a corner placed in the projection of the origin into the patch plane.
const float	kernelS_D (const float d, const float t, const float b, const float s)

Detailed Description

Wendland kernel definition (3D version).

Macro Definition Documentation

_KERNEL_H_INCLUDED_

#define _KERNEL_H_INCLUDED_

iM_PI

#define iM_PI   0.318309886f

\( \frac{1}{\pi} \) value.

M_PI

#define M_PI   3.14159265359f

\( \pi \) value.

Function Documentation

_Omega()

const float _Omega ( const float  a, const float  b  )

inline

Helper function to compute the solid angle of a rectangular patch, with a corner placed in the projection of the origin into the patch plane.

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Parameters

d	Normal distance of the patch to the origin.
a	Width of the rectangular patch.
b	Height of the rectangular patch.

Returns

Equivalent kernel divergent part

See also

kernelS_D

kernelF()

const float kernelF ( const float  q )

inline

The kernel gradient factor \( F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).

The factor \( F \) is defined such that \( \nabla W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) = \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \cdot F \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).

Parameters

q	Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \).

Returns

Kernel amount

kernelS_D()

const float kernelS_D ( const float  d, const float  t, const float  b, const float  s  )

inline

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kernelS_P()

const float kernelS_P ( const float  q )

inline

An equivalent kernel function to compute the Shepard factor using the boundary integral elements instead of the fluid volume.

For practical purposes, the kernel computation is split in 2 parts: The polynomial part, where trucation errors are acceptable, and the divergent part, which requires an analytical solution. This function computes the polynomial part.

The kernel is defined as follows: \( \hat{W} \left(\rho; h\right) = \frac{1}{\rho^d} \int \rho^{d - 1} W \left(\rho; h\right) d\rho \)

Parameters

q	Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \).

Returns

Equivalent kernel polynomial part

See also

kernelS_D

kernelW()

const float kernelW ( const float  q )

inline

The kernel value \( W \left(\mathbf{r_j} - \mathbf{r_i}; h\right) \).

Parameters

q	Normalized distance \( \frac{\mathbf{r_j} - \mathbf{r_i}}{h} \).

Returns

Kernel value.